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Totally multiplicative sequence with a(p) = (p+2)^2 = p^2+4p+4 for prime p.
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%I #12 Sep 20 2020 04:40:53

%S 1,16,25,256,49,400,81,4096,625,784,169,6400,225,1296,1225,65536,361,

%T 10000,441,12544,2025,2704,625,102400,2401,3600,15625,20736,961,19600,

%U 1089,1048576,4225,5776,3969,160000,1521,7056,5625,200704,1849,32400,2025,43264

%N Totally multiplicative sequence with a(p) = (p+2)^2 = p^2+4p+4 for prime p.

%H G. C. Greubel, <a href="/A167358/b167358.txt">Table of n, a(n) for n = 1..1000</a>

%F Multiplicative with a(p^e) = ((p+2)^2)^e. If n = Product p(k)^e(k) then a(n) = Product ((p(k)+2)^2)^e(k). a(n) = A166590(n)^2.

%F Sum_{k>=1} 1/a(k) = Product_{primes p} (1 + 1/(p^2 + 4*p + 3)) = 1.1773018966974266400752906612246691227245078032189833736353235503076639420... - _Vaclav Kotesovec_, Sep 20 2020

%t a[1] = 1; a[n_] := (fi = FactorInteger[n]; Times @@ ((fi[[All, 1]] + 2)^fi[[All, 2]])); Table[a[n]^2, {n, 1, 100}] (* _G. C. Greubel_, Jun 11 2016 *)

%Y Cf. A166590.

%K nonn,mult

%O 1,2

%A _Jaroslav Krizek_, Nov 01 2009